Open Section Shear Flow

Open Section Shear Flow

Open Section Shear Flow: Thin-Wall Shear Flow, Shear Centre and the Boom Idealization

What the tool does

A beam of thin-walled open cross-section (a channel, zee, I, tee, angle, hat, a bent sheet, a stringer) carrying a transverse shear force develops shear stresses in its walls. Because the walls are thin, the shear stress is taken as uniform through the thickness and directed along the wall, and the quantity that is carried from wall to wall is the shear flow q = τ t, a force per unit length of wall. The tool computes q at every point of the section for the shear loads V_x and V_y, finds the shear centre, checks that the flows balance the loads, and draws the distribution. The section is either one of eight library shapes built from a few dimensions or a boom-skin idealization typed row by row.

Where the shear flow comes from

Take a short length δz of beam and cut a wall element of length δs out of it. The bending stress σ_z changes along the beam because the bending moment changes, and the difference between the direct forces on the two faces of the element must be balanced by the shear flows on its edges:

∂q/∂s + t ∂σ_z/∂z = 0.

With the bending stress of the unsymmetrical bending formula and the moment-shear relations ∂M_x/∂z = V_y, ∂M_y/∂z = V_x, the derivative of the stress is linear in the centroidal coordinates x and y, and integrating along the wall from a free edge, where q = 0, gives (Megson eq. 17.14)

q(s) = −k_x ∫₀^s t x ds − k_y ∫₀^s t y ds,   k_x = (V_x I_xx − V_y I_xy) / (I_xx I_yy − I_xy²),   k_y = (V_y I_yy − V_x I_xy) / (I_xx I_yy − I_xy²).

The integrals are the first moments of the material between the free edge and the point, so this is the familiar q = V Q / I written for two load components and a product of inertia. When the section is symmetric about one axis (I_xy = 0) and only V_y acts, it reduces to q = −V_y Q_x / I_xx. The sign convention is Megson's: x to the right, y up, V_x and V_y positive along +x and +y, and q positive in the direction of increasing s. In the tool every wall runs from its node a to its node b, so a positive q flows from a to b and the figure draws the direction with arrows.

Walking a section with branches

An open section is a tree: its walls join at nodes and enclose no cell. On a channel or a zee the walk is a single path from one free edge to the other. On an I, a tee or a hat there are junctions where three walls meet. The tool handles both the same way: for a point on any wall, the material "behind" it is everything on the a side of that wall when the wall is cut, so q arriving at the start of the wall is −k_x Q_y − k_y Q_x of that whole piece, and along the wall the integral is exact because x and y vary linearly with s:

q(s) = q_a − t_D [ k_x (x_a s + Δx s²/2L) + k_y (y_a s + Δy s²/2L) ].

So q is linear on a flange that lies at constant y under V_y and parabolic on a web, with its vertex where the wall crosses the neutral axis. At a junction the flows arriving from the branches sum to zero, which the tool reports as a residual (zero to round-off when the walk is right). The resultants of the wall flows, Σ F_x and Σ F_y, must equal the applied shear; this follows from the equilibrium the derivation started from and is printed as the balance check.

Thin-wall section properties

Each wall is a line of area t L at its mid-point. Its own second moments are t L (Δy)²/12 about x, t L (Δx)²/12 about y and t L Δx Δy/12 for the product, with the transfer terms to the centroid added; the t³ terms (the wall's second moment about its own mid-line) are dropped, which is the thin-wall approximation. Dimensions are taken on the wall mid-lines, so a channel of overall depth H and flange thickness t_f has h = H − t_f. The approximation is good when t is a small fraction of the wall lengths; the test script compares the shear centre of a channel drawn as a thick polygon (the Section Properties tool's warping-function solution) with the thin-wall value and shows the difference shrinking as the thickness halves.

Shear centre

The shear flows have a resultant equal to the applied shear, and that resultant acts along a definite line. The point through which the resultants of V_x and V_y both pass is the shear centre S: a transverse load applied through S bends the beam without twisting it, and a load applied anywhere else is the same load through S plus a torque. S is a property of the geometry alone. The tool finds it from moments: with V_y = 1 and V_x = 0 the moment of all the wall flows about the drawing origin equals x_s; with V_x = 1 it equals −y_s. The moment of a wall's flow about the origin is (x_a Δy − y_a Δx)/L times ∫q ds, because the lever arm of a force along a straight line is the same at every point of the line.

Two closed forms are printed beside the computed value. For a channel with equal flanges the flange flows form a couple whose moment about the web is balanced by the load acting at a distance e from the web mid-line, on the side away from the flanges (Megson Example 17.2 with the thicknesses kept separate):

e = 3 b² t_f / (h t_w + 6 b t_f),   which for t_w = t_f is 3b² / (h + 6b).

For a circular arc of radius r and half-angle φ (a semicircle has φ = 90°), the flow is q = V_y t r² (cos θ − cos φ)/I_xx with I_xx = t r³ (φ − sin φ cos φ), and its moment about the centre of curvature gives the shear centre at

e = 2 r (sin φ − φ cos φ) / (φ − sin φ cos φ),   which for the semicircle is 4r/π ≈ 1.27 r, outside the arc.

Symmetry settles several shapes without calculation: the shear centre of a zee is at its centroid (the section is symmetric about that point), of an angle at the corner and of a tee at the flange-stem junction (every wall passes through that point, so the flows have no moment about it), and of an I on the web at the height h I₁/(I₁ + I₂) above the bottom flange, where I₁ and I₂ are the flanges' second moments about the web. The tool reproduces each of these.

The boom-skin idealization

Aircraft structures are commonly idealized into booms, concentrated areas at the stringers and flanges that carry all the direct stress, joined by skins that carry only shear (Megson Section 20.3; Bruhn A14 and A15). The skins then have a direct-stress thickness t_D = 0, so the integral term vanishes and q is constant between booms, jumping at each boom r by −k_x B_r x_r − k_y B_r y_r (eq. 20.6). The section properties are those of the booms alone. The tool's table takes the booms in order along the section with the thickness of the skin that joins each row to the next; the last row's skin is not used. Megson's Example 20.3, the default case, is a channel of four 300 mm² booms 400 mm apart vertically and 200 mm horizontally under 4.8 kN: I_xx = 4 × 300 × 200² = 48 × 10⁶ mm⁴, the flange skins carry 6 N/mm and the web skin 12 N/mm, whose resultant 12 × 400 = 4,800 N is the applied load, and the shear centre is 2 × 6 × 200 × 200 / 4,800 = 100 mm to the left of the web. The constant web flow is the average of the parabolic flow the same channel would carry with real walls. The box "Skins carry bending" keeps the booms but also counts the skins in the section properties with their full thickness, which is the usual refinement when the skin is a substantial part of the section.

Shear stress, torque and twist

The shear stress in a wall is τ = q/t with the wall's actual thickness, largest where |q| is largest, normally at the neutral axis of the web. When the load line misses the shear centre the section also carries the torque T = (x_V − x_s) V_y − (y_V − y_s) V_x. An open section resists torque poorly: its Saint-Venant constant is J = Σ L t³/3, the twist rate is θ′ = T/(G J) and the torsional shear stress is T t/J, largest in the thickest wall and to be added to the bending shear stress on the same face. Restraint of warping at a built-in end raises the stiffness and adds longitudinal stresses (Timoshenko 1945); the tool neglects it.

Reading the results

Figure 1 draws the section on its wall mid-lines with the shear flow as a shaded diagram along every wall, its height proportional to |q| and its arrows in the direction of flow; a diagram that crosses to the other side of a wall marks a change of flow direction, as on the flanges of a zee. Table 2 lists q at both ends of every wall and the extreme on the wall; an "interior" extreme is the vertex of the parabola. The sign of q in the table is relative to the wall's a → b direction given in the same row. Figure 2, for sections that are a single path, unwraps the walls into a line and plots q against the distance along it. The shear centre is given in the drawing coordinates and as an offset from the centroid, so the lever arm of any load line can be read directly.

Limits

Elastic, thin-wall, straight prismatic beam. Not covered: closed cells (see the Box Beam Shear Flow tool), curved walls other than the circular arc (which is traced as a polygon of one-degree segments, accurate to better than 0.01 %), taper along the beam (which shifts part of the shear into the flanges), shear lag in wide flanges, warping restraint, plasticity, and buckling of the walls under the shear stress found here (see the Diagonal Tension Web and Plate Buckling tools).